Multi-objective modeling and optimization for scheduling of cracking furnace systems☆

Peng Jiang ,Wenli Du ,2,*

1 Key Laboratory of Advanced Control and Optimization for Chemical Process,Ministry of Education,Shanghai 200237,China

2 School of Information Science and Engineering,East China University of Science and Technology,Shanghai 200237,China

1.Introduction

Ethylene is the most important basic raw material in petrochemical industry,and it plays a decisive role for industrial production process.Ethylene cracking furnace is the basic unit in ethylene production,and the tube cracking furnace is occupying the dominant position.In the past,most of the ethylene plants used only a single feed for ethylene production.However,the increasing demand for ethylene and the economic benefit have forced the plants to move toward multiple feeds[1].During the process ofethylene continuous production,carbon produced by side reactions of hydrocarbon pyrolysis will accumulate coke in the internal surface of cracking furnace tube;this can limit the performance of heat transfer of furnace tube and products yield.Carbon can also lead to the increase of tube temperature of pressure drop;this makes it necessary to decoke the cracking furnace regularly.However,ethylene cracking furnace will emit a lot of carbon dioxide,carbon monoxide and dust during the decoking stage.Considering the above factors and target to maximize benefit and minimize the coking amount in the case of fixed feed rate of the ethylene cracking furnace,how to distribute the raw material,processing batches,time and weight the tradeoff between high conversion and high cleanup costs constitute a very complex multi-objective scheduling optimization problem.

The cyclic scheduling of a furnaces system where multiple feeds are simultaneously processed in multiple furnaces run in parallel problem was first proposed by Jain V,Grossmann[2].The authors abstracted this problem into a cyclic scheduling of continuous parallel-process units with exponential decaying functions for product yields,formulated as a mixed-integer nonlinear programming(MINLP)model and solved the problem with the branch and bound algorithm.Based on the previous model,Schulz,Bandoni,and Diaz[3]proposed an extension for ethane-fed ethylene plants with more consideration of recycled ethane as cracking feed.Soon after,a discrete time-based MINLP modelwas developed to study cyclic optimalfurnace shutdowns and downstream separation train system.It employs a time-dependent empirical variable of coil internal roughness as the indicator for furnace shutdown operation.Limet al.[4]proposed a scheduling model by incorporating a neural-network based on simulation data of industrial field and proposed three alternative solution strategies that circumvent the nonlinear terms in the objective function to solve MINLP.Limet al.[5]further proposed a proactive scheduling strategy for the decoking operation of an industrial naphtha cracking furnaces system to deal with the uncertainty from the measurement errors and unexpected changes in the coke growth rate and to determine appropriate rescheduling points before actual operational problems arise.Liu Cet al.[6]considered the optimal allocation of multiple feeds to multiple cracking furnaces to maximize the pro fitability of industrial cracking processes.Meanwhile,the schedule model can inherently avoid simultaneous cleanups for multiple furnaces,and thus allow the scheduling results to be more applicable in reality.Zhaoet al.[7]proposed an MINLP model with multiple feeds by considering the secondary ethane cracking,no simultaneous cleanup constraints.Compared with a heuristic method,their model was proven to have more economic pro fits.Zhaoet al.[8]further developed a new MINLP-based reactive scheduling strategy,which can dynamically generate reschedules based on the new feed deliveries,the leftover feeds,and current furnace operating conditions.Shanget al.[9]considered the mechanistic models cannot be applied online effectively;a yield model was developed for ethylene&propylene based on PSO-SVM and a new MINLP model was developed to obtain feed scheduling strategies for cracking furnace systems.Jian[10]established an ethylene yield model under different operation conditions which combines feedstock and the load changing of each feed and a model of furnace tube coking amount changes with processing time,and then a new scheduling model for furnace system aiming at minimizing the emission of pollutants is raised.

The above-mentioned studies always focused in maximizing the average pro fit of cracking furnace system or other single objective,however,in actual industry production,there are always more than one objective of scheduling problem for ethylene cracking furnace system needs to be optimized.Numerous types of needs existing in the production process,such as pro fit,time,difficulty level of production,pollutants,etc.Meanwhile,some of targets cannot be measured on pro fit,for example,past researchers always build a single objective model to optimize the problem,and they simply count the economic loss of the pollution.However,the damage on environment caused by pollutants cannot be measured by any economic model;it cannot be ignored as an important factor during production process.So it becomes necessary to build a scheduling model which simultaneously takes into account more than one target.A multi-objective mixed integer nonlinear programming(MOMINLP)model was established in this paper,considering both maximizing the average pro fit and minimizing the emission of pollutants which is more suitable in actual production.

The general form of a MOMINLP is shown as Eq.(1).

whereFrepresents thep-dimensional objective function.Equationsh(x,y)andg(x,y)represent equality and inequality constraints.The continuous and discrete decision variables are given byxandy,R,Zrepresent the decision spaces.

The solving method of multi-objective mixed integer nonlinear programming problem can be divided into two categories:mathematical programming method and multi-objective evolution algorithms.At present,mathematical programming are wildly used to solve these problems,the solving strategies are based on the simplified model or convert the problem of multi-objective optimization into a single-objective one,but this method shows little effect on complicated model in actual production.As for the multi-objective evolution algorithms,considerable research has already been undertaken;however,correlative work is seldomdue to the complex computing process of MOMINLP with nonlinear terms and mixed variables[11].Shi and Yao[12]proposed a steady state non-dominated sorting genetic algorithm(SNSGA)which combined the steady state idea of SSGA and the fitness allocation approach of NSAG for MILP and MINLP problem.The algorithm then applied into MOMINLP and obtained Pareto front of the test function,but the convergence and distribution of this algorithm is poor.

MOMNILP problem existed in most of chemical production process.An effective method to solve this kind of problem could have important implications in practical production,and since the complexity of MOMINLP problem,the only previous solution for this problem has a very poor performance.Therefore,a hybrid coding non-dominated sorting genetic algorithm is adopted to solve the multi-objective scheduling problem put forward in this paper.At first,the Pareto frontier of the test function is obtained,the result shows the algorithm can obtain the Pareto frontwith better convergence and distribution,such an algorithm is verified to accelerate convergence process,enhance searching efficiency and solving precision as well as avoid getting into a local optimum.Finally the verification was carried out by means of a simulation on an example of a domestic ethylene plant with the MOMINLP model established in this paper.

2.Scheduling Problem Description

As different materials in different cracking furnace has different product yields,coking rate,product prices and decoking costs,the scheduling of raw materials of the ethylene furnace group emphasizes the optimal distribution of various kinds of raw materials supplied to parallel units in a certain cycle time.The next section describes the necessity of schedule with decoking frequency.

It is generally known that decoking time and frequency is the one of main causes that affects the performance of the ethylene plant.In actual production,the conversion to ethylene is assumed to decrease exponentially with time,therefore,the furnace has to be shut down and cleaned to restart operation at a higher performance level.The time between the cleanings must be determined by considering the trade-off between cleanup costs and the average pro fitability.

Relationship between ethylene yield and decoking frequency is shown as Fig.1.

As seen in Fig.1,when the decoking frequency is higher,the average ethylene yield is higher,butfrequently decoking willdelay the production and increase the decoking costs.Besides,different decoking frequency will lead to different coking amount,when targeting as minimizing the coking amount,clearly there is a trade-off between high product yield,low coking amount and low decoking costs.For the case that single furnace handling single feed,the scheduling modelcan be easily established,but when it comes to the case that multiple feeds processed in multiple furnaces,scheduling problem will be a rather complicated optimization problem.

Fig.1.Relationship between ethylene yield and decoking frequency.

The multi-objective scheduling optimization model of ethylene furnace system proposed in this paper is based on the following prerequisites:

(1)All the product yield of different feeds and the coking amount modelofthe internalsurface in cracking furnace tube are obtained by fitting the simulated data from COILSIMID[10].

(2)In one process batch,a cracking furnace only processes one feed at the same time.

(3)In the process ofscheduling optimization,assuming that the feed supply for all materials are sufficient.

3.Multi-objective Scheduling Model

3.1.Problem statement

In a typical furnaces system as shown in Fig.2,there are usually several different feeds arriving continuously and stored at different charging tanks before cracked in the furnaces.

Fig.2.A typical ethylene cracking furnaces system.

The multi-objective scheduling optimization model contains two objective functions and several constraints,the problem proposed in this paper can be modeled as a multi-objective mathematical programming problem that corresponds to an MOMINLP in which two nonlinear objective functions have to be minimized subject to nonlinear constraints.The objective functions and constraints for the problem are as follows.

3.2.Objective function

The first objective of this problem is to maximize the net pro fit in unit time of ethylene furnace system.The objective function is as follows:

In the above equation,the processing rate of feediin furnacejis donated asDi,j,the price parameter is given byPi,the conversion to ethylene is assumed to decrease exponentially with time as inci,j+ai,je-bi,jt,whereai,j,bi,j,ci,jare given parameters for feediand furnacejthat are typically fitted with plant data.The setup and cleanup cost for the feediin furnacejis given by the constantCsi,j.Total processing time of feediin furnacejis given byti,j,if a feed is not assigned to a furnace,its value is zero.ni,jmeans the number of sub-cycles of feediin furnacej.Its value is zero if feediis not assigned to furnacej,then the length of each sub-cycle(T)is given byti,j/ni,j,whereTcyclemeans the common cycle time for all the furnaces.

The second objective of this problem is minimizing the coking amount in the furnace tube of the ethylene furnace system,which takes into accountthe problems offurnace tube coking amountchanging and pollutants emission.From the above we can see that the production as a function of time can be given by the equation:

3.3.Mass balance

The amount of each feed processed by a furnaces system should be equalto thatofthe corresponding feed thatarrived into the plantduring the total cyclic scheduling horizon.For each cycle of operation there should be no accumulation or depletion of feedstock in the system.The constraint for the feed material is the following:

Since the multi-objective scheduling modelin this paperis established with fixed furnace load,the variableFican be seen as a constant,so the mass constraint can be converted to the following inequalities:

where Fupiand Floirepresent the upper and lower bound of the feeds rate,respectively.

3.4.Integer constraint

The sub-cycle number of processing feediin furnacejis an integer variable in certain limits.

Hereni,jis an integer variable,and the parameterNis selected so as to be consistent with the upper bound on the number of sub-cyclesni,j.

3.5.Timing constraint

There are three constraints for the integrated cyclic problem:

(1)The total time allocated for feediin furnacejshould be equal to the sum of processing time and decoking time for all sub-cycles.

(2)The total time allocated(processing time and decoking time)on any furnace is not more than the cycle time horizon.

(3)The time of feediin furnacejwill be given as zero when the corresponding sub-cycle number is zero.

3.6.Bounds

Apart from the total processing time and sub-cycle number,the remaining variables should have a lower bound of zero.

4.Introduction of MDNSGA-II

In recent years,the research of multi-objective optimization has gradually become a spotlight of research and genetic algorithm(GA)is one of the most effective solutions.It is a random algorithm by means of the nature choice and encourages of genetic populations.The genetic algorithm can handle multiple possible solutions at the same time,pass to operate once and identify the superior solution[13].Overthe pastseveraldecades genetic algorithm was considered especially suited to solve this kind of questions in some sense.The non-dominated sorting genetic algorithm(NSGA-II)adopts the fast non-dominated sorting approach,elitist strategy and a crowded comparison approach,it is verified as one of the most effective algorithm to solve multi-objective problem.NSGA-II has no need for specifying a sharing parameterand low operationalcomplexity.Compared with the traditional optimization algorithm,NSGA-II has some obvious superiority[14].

To solve the MOMINLP modelin this paper,an improved MDNSGA-II with hybrid coding and genetic operators was proposed and the detailed strategies are as follows.

4.1.Hybrid coding strategy

For optimization problem with mixed variables,how to deal with the discrete part of the decision variables is always the key factor.Previous study in this respect have mainly focused on rounding-off strategy[15];this kind ofmethod can solve the problem theoretically,butwhen the optimization problem has a steep peakedness,the solution result would have a large deviation with regard to the actual optimum solution,as shown in Fig.3,there is a large difference between actual result after rounding off the decision variables and optimal value.Besides,the solutions are prone to developing to infeasible solutions after rounding-off[16].Therefore,it is essential that using different encoding method and genetic operator.

Binary,gray,real-number and other coding methods can be adopted in NSGA-II,which can deal with hybrid variables conveniently.As for the ethylene furnace scheduling problem in this paper,due to the fact thatdecision variables in the model has a large range and contain integerand continuous variables,binary coding is too complicated forthis kind of problem,and there are various problems,such as Hamming cliff,computing precision.In view of this,the integer and real number hybrid coding technique was employed in this paper to describe the sub-cycle number and processing time in the cycle scheduling problem.

Fig.3.Shortcoming of rounding-off strategy.

4.2.Hybrid crossover operator

Crossoveris the key operatorin the genetic algorithms;itis the main way to generate new individuals and make the offspring inherit excellent genes from the parents.

4.2.1.Crossover operator for continuous variables

Simulated binary crossover(SBX)operator[17]was adopted for the real-coded variables in the scheduling model.SBX operator is one of the most frequently used crossover operator and has been successfully applied to GAs with real-number coding strategy,it can be described as following:for arbitrary parent individualsx1andx2,the way to generate offspring individualsc2andc2can be expressed as follows:

In the above equation,uis the random number between 0 and 1,η is a constant parameters of the crossing operator.

4.2.2.Crossover operator for discrete variables

As for the discrete variables in the decision variables,due to the ranges of the decision variables are 0–4,crossover operators for the 0–1 variables are not effective for problem in this paper.By comparing the optimization results of different crossover operators,uniform crossover operator was adopted in this paper[18].Uniform crossover uses a fixed mixing ratio between two parents;unlike one-and two-point crossover,uniform crossover enables the parent chromosomes to contribute the gene level rather than the segment level.For the parent individual:x1=a11a12…a1Landx2=a21a22…a2L,the way to generate offspring individuals can be expressed as shown in Fig.4.

Fig.4.Uniform crossover.

Every allele is exchanged between a pair of randomly-selected chromosomes with a certain probabilityp,known as the swapping probability.

4.3.Hybrid mutation operator

Mutation operatoris an essentialpartofgenetic algorithm;itcan enhance the local search ability and population diversity of genetic algorithm and avoid premature convergence by preventing the population of chromosomes from becoming too similar to each other,thus slowing or even stopping evolution.

4.3.1.Mutation operator for the real-coded variables

Gaussian mutation operator with boundary processing was used for the real-coded variables in the scheduling model.Gaussian mutation has many characteristics such as concentricity,symmetry,uniform variability and so on.Gaussian mutation makes small random changes in the individuals in the population.It adds a random number from a Gaussian distribution with mean zero to each vector entry ofan individual.The variance ofthis distribution is determined by the parameters scale and shrink.During mutation operation,this operator adds a unit Gaussian distributed random value with μ mean and σ2variance to the chosen gene.For the parent individual:X=x1x2…xk…xL,the detailed process of Gaussian mutation are as follows:

(1)Initializing a random numberQwith normal distribution;

(2)Replace the chosen genexkfrom parent individuals withQ;

(3)If the new gene is out of the boundary,it will be set as the boundary value.

4.3.2.Mutation operator for discrete variables

Considering thatthe decision variables in the scheduling modelhave a large range,the decision variables would have a big difference after mutation operation,therefore,an improved discrete mutation operator was adopted to handle the integer variables in the model,and the principle is mapping random variables to discrete solution set.Suppose η be a random variable distributed between 0 and 1.For the parent individualX=x1x2…xk…xL,the detailed process of discrete mutation are as follows:

Fig.5.Pareto front of improved MDNSGA-II(a)and SNSGA(b).

In the above equation,INT means rounding operation and εkmeans the minimum difference between discrete variables[15].

4.4.Simulation result

To testthe algorithm performance for multi-objective mixed integer nonlinear optimization problem,a MOMINLP test function presented in literature[19]was selected to verify the algorithm.

This test function contains 3 binary discrete variables and 3 continuous variables,both objectives and constraints are all nonlinear and therefore it is a typical MOMINLP problem.The MDNSGA-II algorithm in this paper was adopted to solve this test function with 100 individuals and 100 iterations;the simulation result is shown in Fig.5.

Compared with the simulation result of SNSGA with 350 iterations and 100 individuals presented in literature[12],it can be seen that the Pareto front obtained by MDNSGA-II in this study has a better distribution and convergence,and the number of iterations are only 100,which means the convergence speed is faster.

To further validate the feasibility of the improved algorithm,two performance metrics have been adopted to evaluate the quality ofthe obtained optimal solutions compared to the Pareto-optimal front obtained MOMDPSO[20],which includes generational distance(GD)and spread(SP).The population size and iteration numbers for these three algorithms are all 100,the repository size and other parameters for MO-MDPSO are the same with those of literature[20].To avoid the one-time occasionality in experiment and ensure the validity of the test,30 independent repeated experiments have been done for each algorithm.

The optimal front of these two algorithms are shown in Fig.6;we can see thatthe Pareto frontofMDNSGA-IIbasically coincided with the Pareto front obtained by MD-MOPSO,and the solution set of MD-MOPSO are partially dominated by the solution set of MDNSGA-II.

Fig.6.Pareto front of two algorithms.

The experiments results of generational distance and spread are shown in Table 1.It's easy to see that GD and SP metrics of improved MDNSGA-II are better than those of MO-MDPSO with the exception of the minimum value of GD.

Table 1Experimental results of test function

In summary,the simulation results show that the algorithm in this paper can find the accurate Pareto front with a better distribution and convergence.

5.Case Study of Furnace System

We selected a domestic ethylene plant as the example;optimal Pareto set was obtained with the multi-objective scheduling model and improved MDNSGA-II presented in this paper and compared with the simulation result of constrained multi-objective particle swarm optimization algorithm.In addition,with the same parameter,the validity of improved MDNSGA-II to solving multi-objective scheduling model established in this paper was validated by comparing the scheduling results between multi-objective optimization model and the one of single-objective model.

The scheduling problem of ethylene furnace system primarily solves three questions:the assignment of various feeds to different furnaces;the number of sub-cycles of each feed on the assigned furnace and the length of these sub-cycles.There are three different types of furnace(GK-VI,GK-V and GBL-III,which are referred to as 1,2 and 3)processing three kinds of feeds(NAP,LNAP,LPG,which are referred to as A,B,C)in this case.Based on the ethylene yield model,a new MOMINLP model aimed atmaximizing benefitand minimizing coking amount ofethylene furnace systemwas established in this paper.Various parameters associated with the model presented in Jian's paper[10]are listed in Table 2.

Table 2Parameter values of domestic ethylene plant case

This article assumes that the overall cycle time for all furnaces is 240 days.Based on the above parameters,a multi-objective model with 2 nonlinear objectives and 9 nonlinear constraints was established,and the decision variables of this scheduling model contains 9 continuous variables and 9 discrete variables.For comparison purposes,SNSGA was first adopted to solve this problem,the particle number and repository size were set to 200,and the number of iterations is 2000 times.The optimization result of this algorithm is shown in Fig.7;it can easily be seen that the constrained SNSGA cannot solve the Pareto front of the model even with a high time complexity,and there are few feasible solutions obtained,the global search ability of SNSGA is poor and it easy to fall into local optimum.

Fig.7.Scheduling result with SNSGA.

Then we apply the improved MDNSGA-II to the multi-objective scheduling model;the population size was 200,and the number of iterations is 1000 times;the feasible solutions obtained by this algorithm were 200.Population's distribution of different generation is also shown in thefig.8;we can see that the solution aggregate tend toward to the optimal Pareto front with the increasing of iterations,the simulation result indicates that improved MDNSGA-II can effectively converge to the Pareto frontier of scheduling model,compared with the result of SNSGA,the advantages of MDNSGA-II is obvious.

Fig.8.Solution sets with the increase of iteration times by MDNSGA-II.

As it shows in Fig.8,A and B represent the extreme value of two scheduling objectives in the Pareto front of the scheduling problem,but the other objective is too small or large;it's not a good option when making decision in actual production.In this paper,we selected three compromise solutionsf1*= （f1，f2）= （0.1619，6.976× 105）,f2*=（f1，f2）= （0.152，6.282 × 105）,f3*= （f1，f2）= （0.247，7.042 × 105）from well distributed part of the Pareto solution set to illustrate the effective of optimization result.The corresponding decision variables,which are also the scheduling results,are shown in Table 3.

Table 3Scheduling results of MDNSGA-II

From the above scheduling results of multi-objective model,we can see the assignment of each feed to different furnaces,the number of sub-cycles of each feed and the length of these sub-cycles.The scheduling order of feedstock in cracking furnaces is shown in Fig.9.

In order to verify the effectiveness of the multi-objective model and algorithm,the scheduling solutions of single-objective model[10]are presented in Table 4.

Finally,we can see that A and B represent the extreme value of two scheduling objectives in the Pareto front of the scheduling problem,a small change of one objective will cause a large variation of another objective near this two region,therefore,it's nota good option when making decision in the actualproduction.In this paper,we choose thefirstsolution as comparison,fromthe above table we can see thatthe average netpro fit of ethylene furnace system is 697598 USD per day obtained from the multi-objective scheduling model established in this paper,and the result ofsingle-objective modeltargeting atmaximizing pro fitis649640 USDper day.Hence,the optimal value is 7.38%higher than the single-objective solution.As for the average coking amount of ethylene furnace system,the optimal result of multi-objective is 0.1619 kg per ton ethylene produced,and the optimal result of single-objective model is 0.l69 kg,the average coking amount has been lowered by about 4.20%with the new model.

6.Conclusions

A multi-objective scheduling problem dealing with the scheduling of multiple feeds on parallel units has been considered in this article;an MOMINLP model and a MDNSGA-II has been developed to determine the cyclic schedules for these types of problems.The results show that the schedules obtained using MDNSGA-II can lead to a better Pareto front of test function;it was also shown that the optimal results in this paper not only improved the pro fit but also reduced the coking amount in the actual production example with up to three feeds and three furnaces.

[1]F.Q.Zhang,G.Y.Bian,China's ethylene industry,"eleventh five-year"retrospect and prospect,Pet.Technol.Forum30(2)(2011)13–17.(in Chinese)

[2]V.Jain,I.E.Grossmann,Cyclic scheduling of continuous parallel-process units with decaying performance,AIChE J.44(7)(1998)1623–1636.

[3]E.P.Schulz,J.A.Bandoni,M.S.Diaz,Optimal shutdown policy for maintenance of cracking furnaces in ethylene plants,Ind.Eng.Chem.Res.45(8)(2006)2748–2757.

[4]H.Lim,J.Choi,M.Realff,Development of optimal decoking scheduling strategies for an industrial naphtha cracking furnace system,Ind.Eng.Chem.Res.45(16)(2006)5738–5747.

[5]H.Lim,J.Choi,M.Realff,Proactive scheduling strategy applied to decoking operations of an industrial naphtha cracking furnace system,Ind.Eng.Chem.Res.48(6)(2009)3024–3032.

[6]C.Liu,J.Zhang,Q.Xu,Cyclic scheduling for best pro fitability of industrial cracking furnace system,Comput.Chem.Eng.34(4)(2010)544–554.

[7]C.Zhao,C.Liu,Q.Xu,Cyclic scheduling for ethylene cracking furnace system with consideration of secondary ethane cracking,Ind.Eng.Chem.Res.49(12)(2010)5765–5774.

[8]C.Zhao,C.Liu,Q.Xu,Dynamic scheduling for ethylene cracking furnace system,Ind.Eng.Chem.Res.50(21)(2011)12026–12040.

[9]B.P.Shang,W.L.Du,Y.K.Jin,F.Qian,Modeling and optimization of scheduling for ethylene cracking furnace systems scheduling with consideration of changing feedstock,CIESC J.64(12)(2013)4312-4312.(in Chinese)

[10]H.D.Jian,Modeling and Optimization of Scheduling for Cracking Furnace System(M.S.Thesis)ECUST,Shanghai,2014(in Chinese).

[11]Y.Jincai,Q.Bo,C.Huanong,Multiobjective optimization of phosgene absorber using NSGA-II,Comput.Appl.Chem.25(2)(2008)181–184.

[12]L.Shi,P.J.Yao,Multi-objective evolutionary algorithms for MILP and MINLP in process synthesis,Chin.J.Chem.Eng.9(2)(2001)173–178.

[13]R.Hassan,B.Cohanim,W.O.De,A comparison of particle swarm optimization and the genetic algorithm,Proceedings of the 1st AIAA Multidisciplinary Design Optimization Specialist Conference 2005,pp.18–21.

[14]K.Deb,A.Pratap,S.Agarwal,A fast and elitist multiobjective genetic algorithm:NSGA-II,IEEE Trans.Evol.Comput.6(2)(2002)182–197.

[15]D.K.He,F.L.Wang,Z.Z.Mao,Study on application of genetic algorithm in discrete variables optimization,J.Syst.Simul.18(5)(2006)1154–1156.

[16]S.Rajeev,C.S.Krishnamoorthy,Discrete optimization of structures using genetic algorithms,J.Struct.Eng.118(5)(1992)1233–1250.

[17]K.Deb,R.B.Agrawal,Simulated binary crossover for continuous search space,Complex Syst.9(2)(1995)115–148.

[18]G.Syswerda,Uniform crossover in genetic algorithms,International Conference on Genetic Algorithms,Morgan Kaufmann Publishers Inc.,San Mateo,CA 1989,pp.2–9.

[19]T.I.Dimkou,K.P.Papalexandri,Aparametric optimization approach for multiobjective engineering problems involving discrete decisions,Comput.Chem.Eng.22(22)(1998)951–954.

[20]W.Tong,S.Chowdhury,A.Messac,A multi-objective mixed-discrete particle swarm optimization with multi-domain diversity preservation,Struct.Multidiscip.Optim.53(3)(2015)1–18.

Table 4Optimal results of multi-objective and single-objective model